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From what I can tell, it seems like he's just taking chord shapes all the way up the piano...

Yes, and the technique used is a version of standard arpeggio technique.

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"If we continually try to force a child to do what he is afraid to do, he will become more timid, and will use his brains and energy, not to explore the unknown, but to find ways to avoid the pressures we put on him." (John Holt)

There are two things that happen in this kind of arpeggio playing. The first is playing a chord as a succession of individual notes. For simplicity, take the C major chord. The fingers you use are 1-2-3-5 on the notes C-E-G-C. Start with practicing just the broken C major chord (slowly at first as Damon suggested), with 1-2-3-1-2-3-5, all without shifting your hand up the keyboard.

The second thing that happens is shifting the hand/arm up the keyboard. So you can practice playing the solid chord, then jumping up an octave to play the chord again, and then jump to the next octave up.

The third thing, is to put 1&2 together - but do this after you've become comfortable with those steps.

Note1: This kind of arpeggio technique does not use what is referred to as "thumb-under", which IMO is used more for slow arpeggios that need to be played connected.

Note2: it's not easy to play as fast as the guy in the video.

So that's the technique, but I have no idea what chords he is playing...

A word of caution to anyone attempting fast arpeggios: Do NOT attempt to connect the 4th or 5th finger with the thumb by turning the thumb under, as this causes unnatural twisting and may lead to injury. Simply shift the hand up the keyboard, and make sure to support this movement with your arm.